Sarine draws a triangle. She measures the length of the sides and records her measurements as follows. What is the measure of angle C of the triangle?

### Law of Cosines with SSS

The Law of Cosines, , can be rearranged to facilitate the calculation of the measure of angle when and are all known lengths.

which can be further manipulated to .

Let's find the measure of the largest angle in the triangle with side lengths 12, 18 and 21.

First, we must determine which angle will be the largest. Recall from Geometry that the longest side is opposite the largest angle. The longest side is 21 so we will let since is the angle we are trying to find. Let and and use the formula to solve for as shown. It doesn’t matter which sides we assign to and . They are interchangeable in the formula.

**Note:** Be careful to put parenthesis around the entire numerator and entire denominator on the calculator to ensure the proper order of operations. Your calculator screen should look like this:

Now let's find the value of , to the nearest degree.

The angle with measure will be angle so and . Remember, and are interchangeable in the formula. Now we can replace the variables with the known measures and solve.

Finally, let's find the , if and .

First, let’s rearrange the formula to reflect the sides given and requested angle:

, now plug in our values

**Examples**

**Example 1**

Earlier, you were asked to find the measure of angle C of the triangle that has sides a = 3, b = 4, and c = 5.

We can use the manipulated Law of Cosines to solve for C.

Therefore, the triangle is a right triangle.

#### Example 2

Find the measure of in the diagram:

#### Example 3

Find the measure of the smallest angle in the triangle with side lengths 47, 54 and 72.

The smallest angle will be opposite the side with length 47, so this will be our in the equation.

#### Example 4

Find , if and .

Rearrange the formula to solve for

### Review

Use the Law of Cosines to find the value of , to the nearest degree, in problems 1 through 6.

- Find the measure of the smallest angle in the triangle with side lengths 150, 165 and 200 meters.
- Find the measure of the largest angle in the triangle with side length 59, 83 and 100 yards.
- Find the if and .
- Find the if and .
- Find the if and .
- A triangular plot of land is bordered by a road, a fence and a creek. If the stretch along the road is 100 meters, the length of the fence is 115 meters and the side along the creek is 90 meters, at what angle do the fence and road meet?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 13.16.